The most important idea is that log⁡b(x)=y\log_b(x)=y means by=xb^y=x, where b>0b>0, b≠1b\ne 1, and x>0x>0. Logarithm rules turn multiplication into addition, division into subtraction, and powers into coefficients. The change of base formula lets you evaluate logs in any base using a calculator.

When solving logarithmic equations, always check that every log argument is positive.

Key Facts

  • The definition of a logarithm is log⁡b(x)=y\log_b(x)=y if and only if by=xb^y=x, where b>0b>0, b≠1b\ne 1, and x>0x>0.
  • The product rule is log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN)=\log_b(M)+\log_b(N) for positive MM and NN.
  • The quotient rule is log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N) for positive MM and NN.
  • The power rule is log⁡b(Mp)=plog⁡b(M)\log_b(M^p)=p\log_b(M) for positive MM.
  • The change of base formula is log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x)=\frac{\log_a(x)}{\log_a(b)}, commonly written as log⁡b(x)=log⁡(x)log⁡(b)\log_b(x)=\frac{\log(x)}{\log(b)}.
  • Common logarithms use base 1010, so log⁡(x)=log⁡10(x)\log(x)=\log_{10}(x).
  • Natural logarithms use base ee, so ln⁡(x)=log⁡e(x)\ln(x)=\log_e(x).
  • Inverse identities include log⁡b(bx)=x\log_b(b^x)=x and blog⁡b(x)=xb^{\log_b(x)}=x for valid values of bb and xx.

Vocabulary

Logarithm
A logarithm is the exponent needed to raise a base to a given positive number.
Base
The base is the positive number bb in log⁡b(x)\log_b(x), where b≠1b\ne 1.
Argument
The argument is the input xx in log⁡b(x)\log_b(x), and it must be positive.
Common Logarithm
A common logarithm is a logarithm with base 1010, written as log⁡(x)\log(x).
Natural Logarithm
A natural logarithm is a logarithm with base ee, written as ln⁡(x)\ln(x).
Change of Base
Change of base is the formula log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x)=\frac{\log_a(x)}{\log_a(b)} used to rewrite a logarithm in a different base.

Common Mistakes to Avoid

  • Adding logs incorrectly, such as writing log⁡b(M)+log⁡b(N)=log⁡b(M+N)\log_b(M)+\log_b(N)=\log_b(M+N), is wrong because the product rule gives log⁡b(MN)\log_b(MN) instead.
  • Dropping the domain restriction is a mistake because log⁡b(x)\log_b(x) is only defined for x>0x>0 in real-number algebra.
  • Using the power rule backward incorrectly, such as changing log⁡b(Mp)\log_b(M^p) into (log⁡b(M))p\left(\log_b(M)\right)^p, is wrong because the exponent becomes a multiplier, plog⁡b(M)p\log_b(M).
  • Forgetting to check solutions can produce extraneous answers because solving may create values that make a log argument zero or negative.
  • Confusing the base and the argument in exponential form is wrong because log⁡b(x)=y\log_b(x)=y converts to by=xb^y=x, not xy=bx^y=b.

Practice Questions

  1. 1 Rewrite log⁡3(81)=4\log_3(81)=4 in exponential form.
  2. 2 Evaluate log⁡2(32)\log_2(32).
  3. 3 Use log rules to expand log⁡5(x3y25)\log_5\left(\frac{x^3y}{25}\right), assuming x>0x>0 and y>0y>0.
  4. 4 Explain why the equation log⁡4(x−2)+log⁡4(x+2)=1\log_4(x-2)+\log_4(x+2)=1 requires checking the domain before accepting a solution.

Understanding Logarithms Quick Reference

Logarithms are useful because many real quantities change by multiplication rather than by equal additions. A sound level, earthquake measurement, acidity value, and brightness measurement can cover an enormous range. A logarithmic scale compresses that range into manageable numbers.

An increase of one unit on such a scale often represents a fixed multiplication in the original quantity, not a fixed extra amount. This is why a small change in a logarithmic measurement can represent a large physical change. Students should always ask whether a scale is linear or logarithmic before comparing values.

The graph of a logarithmic function has a distinctive shape. It passes through the point where the input is one and the output is zero. It moves slowly to the right because larger outputs require repeated multiplication of the base.

It approaches the vertical axis but never reaches or crosses it. That behavior comes from the fact that an exponential expression with a positive base cannot produce zero or a negative result. For bases greater than one, the graph rises from left to right.

For bases between zero and one, it falls from left to right. These two cases are easy to confuse, so sketching a few points is often more reliable than trying to memorize the shape.

When solving equations with logarithms, the main task is usually to remove the logarithm by using its inverse exponential operation. If two logarithms with the same base are equal, their arguments can be set equal, but only after confirming both arguments are allowed. If several logarithms appear on one side, rules can combine them before converting to exponential form.

A common error is treating the logarithm of a sum as the sum of two logarithms. That rule does not exist.

For example, the logarithm of x plus two cannot be split into the logarithm of x plus the logarithm of two. Parentheses matter because they show the full argument of each logarithm.

Calculators usually provide buttons for common logarithms and natural logarithms, even when a problem uses another base. The change of base method works because dividing two logarithms measures the same exponent in a different reference system. Natural logarithms appear often in science because the number called e describes continuous growth and decay.

They are used in models for cooling, radioactive decay, compound interest, population change, and charging circuits. In word problems, identify what is growing or shrinking by a constant factor over equal time intervals. That clue often points to an exponential model, with logarithms used later to find an unknown time, rate, or exponent.